Abstract

In the first part of this paper we presented a tomographic method to reconstruct the refractive index profile of spherically symmetrical lenses. Here we perform the generalization to lenses that are rotationally symmetrical around the optical axis, as is the ideal human lens. Analysis of the accuracy and versatility of this method is carried out by performing numerical simulations in which different magnitudes of experimental errors and two extreme case scenarios for the likely shape of the refractive index distribution of the human lens are considered. Finally, experimental results for a porcine lens are shown. Conceptually simple and computationally swift, this method could prove to be a valuable tool for the accurate retrieval of the gradient index of a broad spectrum of rotationally symmetrical crystalline lenses.

© 2006 Optical Society of America

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References

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  1. P. L. Chu, "Nondestructive measurements of index profile of an optical fibre preform," Electron. Lett. 13, 736-738 (1977).
    [CrossRef]
  2. D. Marcuse, "Refractive index determination by the focusing method," Appl. Opt. 18, 9-13 (1979).
    [CrossRef] [PubMed]
  3. V. I. Vlad and N. Ionescu-Pallas, "New treatment of the focusing method and tomography of the refractive index distribution of inhomogeneous optical components," Opt. Eng. (Bellingham) 35, 1305-1310 (1996).
    [CrossRef]
  4. D. Y. C. Chan, J. P. Ennis, B. K. Pierscionek, and G. Smith, "Determination of the 3-D gradient refractive indices in crystalline lenses," Appl. Opt. 27, 926-931 (1988).
    [CrossRef] [PubMed]
  5. B. K. Pierscionek and D. Y. C. Chan, "Refractive index gradient of human lenses," Optom. Vision Sci. 66, 822-829 (1989).
    [CrossRef]
  6. L. F. Garner, G. Smith, S. Yao, and R. C. Augusteyn, "Gradient refractive index of the crystalline lens of the Black Oreo Dory (Allocyttus Niger): comparison of magnetic resonance imaging (MRI) and laser ray-trace methods," Vision Res.41, 973-979 (2001).
    [CrossRef] [PubMed]
  7. E. Acosta, R. Flores, D. Vazquez, S. Rios, L. F. Garner, and G. Smith, "Tomographic method for measurement of the refractive index profile of optical fibre preforms and rod GRIN lenses," Jpn. J. Appl. Phys., Part 1 41, 4821-4824 (2002).
    [CrossRef]
  8. E. Acosta, D. Vazquez, L. Garner, and G. Smith, "Tomographic method for measurement of the gradient refractive index of the crystalline lens. I. The spherical fish lens," J. Opt. Soc. Am. A 22, 424-433 (2005).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  13. K. F. Barrell and C. Pask, "Nondestructive index profile measurements of noncircular optical fibre preforms," Opt. Commun. 27, 230-234 (1978).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  27. R. C. Gonzalez and R. E. Woods, Digital Image Processing (Addison-Wesley, 1993), Chap. 7, pp. 416-423.
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    [CrossRef] [PubMed]
  30. C. E. Jones and J. M. Pope, "Measuring optical properties of an eye lens using magnetic resonance imaging," Magn. Reson. Imaging 22, 211-220 (2004).
    [CrossRef] [PubMed]
  31. S. Barbero, A. Glasser, C. Clark, and S. Marcos, "Accuracy and possibilities for evaluating the lens gradient-index using a ray tracing tomography global optimization strategy," Invest. Ophthalmol. Visual Sci. 45 Suppl. 1, U706 1723 (2004).

2005 (1)

2004 (2)

C. E. Jones and J. M. Pope, "Measuring optical properties of an eye lens using magnetic resonance imaging," Magn. Reson. Imaging 22, 211-220 (2004).
[CrossRef] [PubMed]

S. Barbero, A. Glasser, C. Clark, and S. Marcos, "Accuracy and possibilities for evaluating the lens gradient-index using a ray tracing tomography global optimization strategy," Invest. Ophthalmol. Visual Sci. 45 Suppl. 1, U706 1723 (2004).

2002 (3)

B. A. Moffat, D. A. Atchison, and J. M. Pope, "Age-related changes in refractive index distribution and power of the human lens as measured by magnetic resonance micro-imaging in vitro," Vision Res. 42, 1683-1693 (2002).
[CrossRef] [PubMed]

E. Acosta, R. Flores, D. Vazquez, S. Rios, L. F. Garner, and G. Smith, "Tomographic method for measurement of the refractive index profile of optical fibre preforms and rod GRIN lenses," Jpn. J. Appl. Phys., Part 1 41, 4821-4824 (2002).
[CrossRef]

J. F. Koretz, C. A. Cook, and P. L. Kaufman, "Aging of the human lens: changes in lens shape upon accommodation and with accommodative loss," J. Opt. Soc. Am. A 19, 144-151 (2002).
[CrossRef]

2001 (1)

L. F. Garner, G. Smith, S. Yao, and R. C. Augusteyn, "Gradient refractive index of the crystalline lens of the Black Oreo Dory (Allocyttus Niger): comparison of magnetic resonance imaging (MRI) and laser ray-trace methods," Vision Res.41, 973-979 (2001).
[CrossRef] [PubMed]

1998 (1)

1997 (1)

1996 (2)

J. R. Kuszak, K. L. Peterson, and H. G. Brown, "Electron microscopy observations of the crystalline lens," Microsc. Res. Tech. 33, 441-479 (1996).
[CrossRef] [PubMed]

V. I. Vlad and N. Ionescu-Pallas, "New treatment of the focusing method and tomography of the refractive index distribution of inhomogeneous optical components," Opt. Eng. (Bellingham) 35, 1305-1310 (1996).
[CrossRef]

1992 (1)

W. S. Jagger, "The optics of the spherical fish lens," Vision Res. 32, 1271-1284 (1992).
[CrossRef] [PubMed]

1991 (1)

G. Smith, B. K. Pierscionek, and D. A. Atchison, "The optical modelling of the human lens," Ophthalmic Physiol. Opt. 11, 359-369 (1991).
[CrossRef] [PubMed]

1990 (1)

W. S. Jagger, "The refractive structure and optical properties of the isolated crystalline lens of the cat," Vision Res. 30, 723-738 (1990).
[CrossRef] [PubMed]

1989 (1)

B. K. Pierscionek and D. Y. C. Chan, "Refractive index gradient of human lenses," Optom. Vision Sci. 66, 822-829 (1989).
[CrossRef]

1988 (1)

1984 (1)

M. C. W. Campbell, "Measurement of refractive index in an intact crystalline lens," Vision Res. 24, 409-415 (1984).
[CrossRef] [PubMed]

1982 (1)

1980 (1)

1979 (1)

1978 (1)

K. F. Barrell and C. Pask, "Nondestructive index profile measurements of noncircular optical fibre preforms," Opt. Commun. 27, 230-234 (1978).
[CrossRef]

1977 (1)

P. L. Chu, "Nondestructive measurements of index profile of an optical fibre preform," Electron. Lett. 13, 736-738 (1977).
[CrossRef]

1971 (1)

1968 (1)

Acosta, E.

E. Acosta, D. Vazquez, L. Garner, and G. Smith, "Tomographic method for measurement of the gradient refractive index of the crystalline lens. I. The spherical fish lens," J. Opt. Soc. Am. A 22, 424-433 (2005).
[CrossRef]

E. Acosta, R. Flores, D. Vazquez, S. Rios, L. F. Garner, and G. Smith, "Tomographic method for measurement of the refractive index profile of optical fibre preforms and rod GRIN lenses," Jpn. J. Appl. Phys., Part 1 41, 4821-4824 (2002).
[CrossRef]

Atchison, D. A.

B. A. Moffat, D. A. Atchison, and J. M. Pope, "Age-related changes in refractive index distribution and power of the human lens as measured by magnetic resonance micro-imaging in vitro," Vision Res. 42, 1683-1693 (2002).
[CrossRef] [PubMed]

G. Smith, B. K. Pierscionek, and D. A. Atchison, "The optical modelling of the human lens," Ophthalmic Physiol. Opt. 11, 359-369 (1991).
[CrossRef] [PubMed]

Augusteyn, R. C.

L. F. Garner, G. Smith, S. Yao, and R. C. Augusteyn, "Gradient refractive index of the crystalline lens of the Black Oreo Dory (Allocyttus Niger): comparison of magnetic resonance imaging (MRI) and laser ray-trace methods," Vision Res.41, 973-979 (2001).
[CrossRef] [PubMed]

Barbero, S.

S. Barbero, A. Glasser, C. Clark, and S. Marcos, "Accuracy and possibilities for evaluating the lens gradient-index using a ray tracing tomography global optimization strategy," Invest. Ophthalmol. Visual Sci. 45 Suppl. 1, U706 1723 (2004).

Barrell, K. F.

K. F. Barrell and C. Pask, "Nondestructive index profile measurements of noncircular optical fibre preforms," Opt. Commun. 27, 230-234 (1978).
[CrossRef]

Beliakov, G.

Bevington, P. R.

P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, 1992), Chap. 11, pp. 205-209.

Blaker, J. W.

Brennan, N. A.

Brown, H. G.

J. R. Kuszak, K. L. Peterson, and H. G. Brown, "Electron microscopy observations of the crystalline lens," Microsc. Res. Tech. 33, 441-479 (1996).
[CrossRef] [PubMed]

Campbell, M. C. W.

M. C. W. Campbell, "Measurement of refractive index in an intact crystalline lens," Vision Res. 24, 409-415 (1984).
[CrossRef] [PubMed]

Chan, D. Y. C.

Chu, P. L.

P. L. Chu, "Nondestructive measurements of index profile of an optical fibre preform," Electron. Lett. 13, 736-738 (1977).
[CrossRef]

Clark, C.

S. Barbero, A. Glasser, C. Clark, and S. Marcos, "Accuracy and possibilities for evaluating the lens gradient-index using a ray tracing tomography global optimization strategy," Invest. Ophthalmol. Visual Sci. 45 Suppl. 1, U706 1723 (2004).

Cook, C. A.

Ennis, J. P.

Fernald, R. D.

R. D. Fernald, "The optical system of the fish" in The Visual System of Fish, R.H.Douglas and M.B. A.Djamgoz, eds. (Chapman & Hall, 1990), pp. 50-52.

Flores, R.

E. Acosta, R. Flores, D. Vazquez, S. Rios, L. F. Garner, and G. Smith, "Tomographic method for measurement of the refractive index profile of optical fibre preforms and rod GRIN lenses," Jpn. J. Appl. Phys., Part 1 41, 4821-4824 (2002).
[CrossRef]

Flusser, J.

R. Halír and J. Flusser, "Numerically stable direct least squares fitting of ellipses," in Proceedings of the Sixth International Conference in Central Europe on Computer Graphics and Visualization (WSCG'98), V.Skala, ed. (Vydavatelstvi Zapadoceske Univerzity, , 1998), Vol. 1, pp. 125-132.

Garner, L.

Garner, L. F.

E. Acosta, R. Flores, D. Vazquez, S. Rios, L. F. Garner, and G. Smith, "Tomographic method for measurement of the refractive index profile of optical fibre preforms and rod GRIN lenses," Jpn. J. Appl. Phys., Part 1 41, 4821-4824 (2002).
[CrossRef]

L. F. Garner, G. Smith, S. Yao, and R. C. Augusteyn, "Gradient refractive index of the crystalline lens of the Black Oreo Dory (Allocyttus Niger): comparison of magnetic resonance imaging (MRI) and laser ray-trace methods," Vision Res.41, 973-979 (2001).
[CrossRef] [PubMed]

Ghatak, A. K.

Glasser, A.

S. Barbero, A. Glasser, C. Clark, and S. Marcos, "Accuracy and possibilities for evaluating the lens gradient-index using a ray tracing tomography global optimization strategy," Invest. Ophthalmol. Visual Sci. 45 Suppl. 1, U706 1723 (2004).

Gonzalez, R. C.

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Addison-Wesley, 1993), Chap. 7, pp. 416-423.

Gullstrand, A.

A. Gullstrand, Appendix II in Helmholtz'sHandbuch der Physiologischen Optik, Vol. 1, 3rd ed. (English translation edited by J.P.Southall, Optical Society of America, Dover, 1962), pp. 351-352.

Halír, R.

R. Halír and J. Flusser, "Numerically stable direct least squares fitting of ellipses," in Proceedings of the Sixth International Conference in Central Europe on Computer Graphics and Visualization (WSCG'98), V.Skala, ed. (Vydavatelstvi Zapadoceske Univerzity, , 1998), Vol. 1, pp. 125-132.

Ionescu-Pallas, N.

V. I. Vlad and N. Ionescu-Pallas, "New treatment of the focusing method and tomography of the refractive index distribution of inhomogeneous optical components," Opt. Eng. (Bellingham) 35, 1305-1310 (1996).
[CrossRef]

Jagger, W. S.

W. S. Jagger, "The optics of the spherical fish lens," Vision Res. 32, 1271-1284 (1992).
[CrossRef] [PubMed]

W. S. Jagger, "The refractive structure and optical properties of the isolated crystalline lens of the cat," Vision Res. 30, 723-738 (1990).
[CrossRef] [PubMed]

Jones, C. E.

C. E. Jones and J. M. Pope, "Measuring optical properties of an eye lens using magnetic resonance imaging," Magn. Reson. Imaging 22, 211-220 (2004).
[CrossRef] [PubMed]

Kaufman, P. L.

Koretz, J. F.

Kumar, D. V.

Kuszak, J. R.

J. R. Kuszak, K. L. Peterson, and H. G. Brown, "Electron microscopy observations of the crystalline lens," Microsc. Res. Tech. 33, 441-479 (1996).
[CrossRef] [PubMed]

Liou, H. L.

Marchand, E. W.

E. W. Marchand, Gradient Index Optics (Academic Press, 1978), Chap. 9, p. 99.

Marcos, S.

S. Barbero, A. Glasser, C. Clark, and S. Marcos, "Accuracy and possibilities for evaluating the lens gradient-index using a ray tracing tomography global optimization strategy," Invest. Ophthalmol. Visual Sci. 45 Suppl. 1, U706 1723 (2004).

Marcuse, D.

Moffat, B. A.

B. A. Moffat, D. A. Atchison, and J. M. Pope, "Age-related changes in refractive index distribution and power of the human lens as measured by magnetic resonance micro-imaging in vitro," Vision Res. 42, 1683-1693 (2002).
[CrossRef] [PubMed]

Montagnino, L.

Moore, D. T.

Pask, C.

K. F. Barrell and C. Pask, "Nondestructive index profile measurements of noncircular optical fibre preforms," Opt. Commun. 27, 230-234 (1978).
[CrossRef]

Peterson, K. L.

J. R. Kuszak, K. L. Peterson, and H. G. Brown, "Electron microscopy observations of the crystalline lens," Microsc. Res. Tech. 33, 441-479 (1996).
[CrossRef] [PubMed]

Pierscionek, B. K.

G. Smith, B. K. Pierscionek, and D. A. Atchison, "The optical modelling of the human lens," Ophthalmic Physiol. Opt. 11, 359-369 (1991).
[CrossRef] [PubMed]

B. K. Pierscionek and D. Y. C. Chan, "Refractive index gradient of human lenses," Optom. Vision Sci. 66, 822-829 (1989).
[CrossRef]

D. Y. C. Chan, J. P. Ennis, B. K. Pierscionek, and G. Smith, "Determination of the 3-D gradient refractive indices in crystalline lenses," Appl. Opt. 27, 926-931 (1988).
[CrossRef] [PubMed]

Pope, J. M.

C. E. Jones and J. M. Pope, "Measuring optical properties of an eye lens using magnetic resonance imaging," Magn. Reson. Imaging 22, 211-220 (2004).
[CrossRef] [PubMed]

B. A. Moffat, D. A. Atchison, and J. M. Pope, "Age-related changes in refractive index distribution and power of the human lens as measured by magnetic resonance micro-imaging in vitro," Vision Res. 42, 1683-1693 (2002).
[CrossRef] [PubMed]

Rios, S.

E. Acosta, R. Flores, D. Vazquez, S. Rios, L. F. Garner, and G. Smith, "Tomographic method for measurement of the refractive index profile of optical fibre preforms and rod GRIN lenses," Jpn. J. Appl. Phys., Part 1 41, 4821-4824 (2002).
[CrossRef]

Robinson, D. K.

P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, 1992), Chap. 11, pp. 205-209.

Sharma, A.

Sivak, J. G.

J. G. Sivak, "Optical variability of the fish lens," in Ref. , pp. 63-74.

Smith, G.

E. Acosta, D. Vazquez, L. Garner, and G. Smith, "Tomographic method for measurement of the gradient refractive index of the crystalline lens. I. The spherical fish lens," J. Opt. Soc. Am. A 22, 424-433 (2005).
[CrossRef]

E. Acosta, R. Flores, D. Vazquez, S. Rios, L. F. Garner, and G. Smith, "Tomographic method for measurement of the refractive index profile of optical fibre preforms and rod GRIN lenses," Jpn. J. Appl. Phys., Part 1 41, 4821-4824 (2002).
[CrossRef]

L. F. Garner, G. Smith, S. Yao, and R. C. Augusteyn, "Gradient refractive index of the crystalline lens of the Black Oreo Dory (Allocyttus Niger): comparison of magnetic resonance imaging (MRI) and laser ray-trace methods," Vision Res.41, 973-979 (2001).
[CrossRef] [PubMed]

G. Smith, B. K. Pierscionek, and D. A. Atchison, "The optical modelling of the human lens," Ophthalmic Physiol. Opt. 11, 359-369 (1991).
[CrossRef] [PubMed]

D. Y. C. Chan, J. P. Ennis, B. K. Pierscionek, and G. Smith, "Determination of the 3-D gradient refractive indices in crystalline lenses," Appl. Opt. 27, 926-931 (1988).
[CrossRef] [PubMed]

Vazquez, D.

E. Acosta, D. Vazquez, L. Garner, and G. Smith, "Tomographic method for measurement of the gradient refractive index of the crystalline lens. I. The spherical fish lens," J. Opt. Soc. Am. A 22, 424-433 (2005).
[CrossRef]

E. Acosta, R. Flores, D. Vazquez, S. Rios, L. F. Garner, and G. Smith, "Tomographic method for measurement of the refractive index profile of optical fibre preforms and rod GRIN lenses," Jpn. J. Appl. Phys., Part 1 41, 4821-4824 (2002).
[CrossRef]

Vlad, V. I.

V. I. Vlad and N. Ionescu-Pallas, "New treatment of the focusing method and tomography of the refractive index distribution of inhomogeneous optical components," Opt. Eng. (Bellingham) 35, 1305-1310 (1996).
[CrossRef]

Woods, R. E.

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Addison-Wesley, 1993), Chap. 7, pp. 416-423.

Yao, S.

L. F. Garner, G. Smith, S. Yao, and R. C. Augusteyn, "Gradient refractive index of the crystalline lens of the Black Oreo Dory (Allocyttus Niger): comparison of magnetic resonance imaging (MRI) and laser ray-trace methods," Vision Res.41, 973-979 (2001).
[CrossRef] [PubMed]

Appl. Opt. (4)

Electron. Lett. (1)

P. L. Chu, "Nondestructive measurements of index profile of an optical fibre preform," Electron. Lett. 13, 736-738 (1977).
[CrossRef]

Invest. Ophthalmol. Visual Sci. (1)

S. Barbero, A. Glasser, C. Clark, and S. Marcos, "Accuracy and possibilities for evaluating the lens gradient-index using a ray tracing tomography global optimization strategy," Invest. Ophthalmol. Visual Sci. 45 Suppl. 1, U706 1723 (2004).

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (3)

Jpn. J. Appl. Phys., Part 1 (1)

E. Acosta, R. Flores, D. Vazquez, S. Rios, L. F. Garner, and G. Smith, "Tomographic method for measurement of the refractive index profile of optical fibre preforms and rod GRIN lenses," Jpn. J. Appl. Phys., Part 1 41, 4821-4824 (2002).
[CrossRef]

Magn. Reson. Imaging (1)

C. E. Jones and J. M. Pope, "Measuring optical properties of an eye lens using magnetic resonance imaging," Magn. Reson. Imaging 22, 211-220 (2004).
[CrossRef] [PubMed]

Microsc. Res. Tech. (1)

J. R. Kuszak, K. L. Peterson, and H. G. Brown, "Electron microscopy observations of the crystalline lens," Microsc. Res. Tech. 33, 441-479 (1996).
[CrossRef] [PubMed]

Ophthalmic Physiol. Opt. (1)

G. Smith, B. K. Pierscionek, and D. A. Atchison, "The optical modelling of the human lens," Ophthalmic Physiol. Opt. 11, 359-369 (1991).
[CrossRef] [PubMed]

Opt. Commun. (1)

K. F. Barrell and C. Pask, "Nondestructive index profile measurements of noncircular optical fibre preforms," Opt. Commun. 27, 230-234 (1978).
[CrossRef]

Opt. Eng. (Bellingham) (1)

V. I. Vlad and N. Ionescu-Pallas, "New treatment of the focusing method and tomography of the refractive index distribution of inhomogeneous optical components," Opt. Eng. (Bellingham) 35, 1305-1310 (1996).
[CrossRef]

Optom. Vision Sci. (1)

B. K. Pierscionek and D. Y. C. Chan, "Refractive index gradient of human lenses," Optom. Vision Sci. 66, 822-829 (1989).
[CrossRef]

Vision Res. (5)

L. F. Garner, G. Smith, S. Yao, and R. C. Augusteyn, "Gradient refractive index of the crystalline lens of the Black Oreo Dory (Allocyttus Niger): comparison of magnetic resonance imaging (MRI) and laser ray-trace methods," Vision Res.41, 973-979 (2001).
[CrossRef] [PubMed]

W. S. Jagger, "The optics of the spherical fish lens," Vision Res. 32, 1271-1284 (1992).
[CrossRef] [PubMed]

M. C. W. Campbell, "Measurement of refractive index in an intact crystalline lens," Vision Res. 24, 409-415 (1984).
[CrossRef] [PubMed]

W. S. Jagger, "The refractive structure and optical properties of the isolated crystalline lens of the cat," Vision Res. 30, 723-738 (1990).
[CrossRef] [PubMed]

B. A. Moffat, D. A. Atchison, and J. M. Pope, "Age-related changes in refractive index distribution and power of the human lens as measured by magnetic resonance micro-imaging in vitro," Vision Res. 42, 1683-1693 (2002).
[CrossRef] [PubMed]

Other (7)

A. Gullstrand, Appendix II in Helmholtz'sHandbuch der Physiologischen Optik, Vol. 1, 3rd ed. (English translation edited by J.P.Southall, Optical Society of America, Dover, 1962), pp. 351-352.

E. W. Marchand, Gradient Index Optics (Academic Press, 1978), Chap. 9, p. 99.

P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, 1992), Chap. 11, pp. 205-209.

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Addison-Wesley, 1993), Chap. 7, pp. 416-423.

R. Halír and J. Flusser, "Numerically stable direct least squares fitting of ellipses," in Proceedings of the Sixth International Conference in Central Europe on Computer Graphics and Visualization (WSCG'98), V.Skala, ed. (Vydavatelstvi Zapadoceske Univerzity, , 1998), Vol. 1, pp. 125-132.

J. G. Sivak, "Optical variability of the fish lens," in Ref. , pp. 63-74.

R. D. Fernald, "The optical system of the fish" in The Visual System of Fish, R.H.Douglas and M.B. A.Djamgoz, eds. (Chapman & Hall, 1990), pp. 50-52.

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Figures (18)

Fig. 1
Fig. 1

Sagittal plane of a crystalline lens with a refractive index distribution n ( x , z ) traversed by a collection of incoming parallel rays at an angle θ to the Z axis.

Fig. 2
Fig. 2

Interpolation of two points, B p r 1 and C p r 1 , in the iterative procedure to obtain a trajectory (dashed curve) approximated to the actual ray path (solid curve) from B p r to C p r .

Fig. 3
Fig. 3

Flow chart of the tomographic algorithm for the monopolynomial reconstruction of the gradient index.

Fig. 4
Fig. 4

Flow chart of the tomographic algorithm for the bipolynomial reconstruction of the gradient index.

Fig. 5
Fig. 5

(a) Profile at x = 0 of the weak (solid curve) and strong (dashed-dotted for the anterior segment, dashed for the posterior) gradient indices, where z shows the distance from z H to the periphery. (b) Profile at z H of the weak (solid curve) and strong (dashed curve) gradient indices.

Fig. 6
Fig. 6

Evolution of the tomographic retrieval for the weak (circles) and strong (squares for the monopolynomial retrieval, stars for the bipolynomial) gradient indices.

Fig. 7
Fig. 7

Distributions of normalized probability of the (a) peak–valley and (b) rms difference between the retrieved and theoretical gradient indices for the particular case of the strong gradient with an added Gaussian error of σ = λ 5 , projections up to 40°, and 500 incoming rays equally spaced within an aperture of 90% of the diameter of the lens. Lower (L) and upper (U) bounds include 68 % of the cases about the mode.

Fig. 8
Fig. 8

Peak–valley difference between the retrieved and theoretical weak gradient indices for σ = λ (circles), σ = λ 5 (stars), σ = λ 10 (squares), σ = λ 20 (diamonds), and σ = 0 (triangles). The error bars represent L and U. (a) and (b) are the results for 500 and 100 incoming rays, respectively.

Fig. 9
Fig. 9

Rms difference between the retrieved and theoretical weak gradient indices for σ = λ (circles), σ = λ 5 (stars), σ = λ 10 (squares), σ = λ 20 (diamonds), and σ = 0 (triangles). The error bars represent L and U. (a) and (b) are the results for 500 and 100 incoming rays, respectively.

Fig. 10
Fig. 10

Peak–valley difference between the retrieved and theoretical strong gradient indices for σ = λ (circles), σ = λ 5 (stars), σ = λ 10 (squares), σ = λ 20 (diamonds), and σ = 0 (triangles). The error bars represent L and U. (a) and (b) are the results for 500 and 100 incoming rays, respectively.

Fig. 11
Fig. 11

Rms difference between the retrieved and theoretical strong gradient indices for σ = λ (circles), σ = λ 5 (stars), σ = λ 10 (squares), σ = λ 20 (diamonds), and σ = 0 (triangles). The error bars represent L and U. (a) and (b) are the results for 500 and 100 incoming rays, respectively.

Fig. 12
Fig. 12

Experimental setup for the tomographic retrieval of an eye lens gradient index as viewed from the measurement camera.

Fig. 13
Fig. 13

Image of the lens surfaces when illuminated with a white LED backlight.

Fig. 14
Fig. 14

Experimental (circles) and theoretical (solid curve) sine of the exit angles at z = t for the spherical homogeneous lens at θ = 0 ° .

Fig. 15
Fig. 15

Map of the isoindicial lines of the retrieved monopolynomial refractive index distribution. The thicker upper curve represents the anterior surface, the thick lower curve the posterior surface.

Fig. 16
Fig. 16

Map of the isoindicial lines of the retrieved bipolynomial refractive index distribution.

Fig. 17
Fig. 17

Experimental (circles) and reproduced (solid curve) sine of the exit angles at different entry heights for the projection at θ 2 = 1.86 ° for the (a) monopolynomial and (b) bipolynomial retrieved gradient indices.

Fig. 18
Fig. 18

(a) Sagittal and (b) equatorial profiles of the monopolynomial (solid curve) and bipolynomial (dashed curve) retrieved gradient indices.

Tables (4)

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Table 1 Coefficients n i j of the Weak (Monopolynomial) and Strong (Bipolynomial) Gradients for Fourth-Order Polynomials

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Table 2 Results of the Tomographic Retrieval for the Weak and Strong Gradients When Using Data from 0°, 20°, and 40° Projections

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Table 3 Accuracy of the Monopolynominal and Bipolynomial Retrievals of the Porcine Lens Gradient Index

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Table 4 Retrieved Coefficients n i j , Eccentricities, and z H of the Monopolynomial and Bipolynomial Refractive Index Distributions of the Porcine Lens

Equations (25)

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n ext sin α ( x , t ) = L ̂ ( x , t ) x .
L ̂ ( x , t ) = n ext 0 x sin α ( x , t ) d x + L ̂ ( 0 , t ) .
S ( A D ) = L ̂ ( D ) L ̂ ( A ) ,
S ( A o D o ) = L ̂ ( D o ) L ̂ ( A o ) ,
L ̂ ( A ) = L ̂ ( A o ) ,
S ( A D ) = S ̃ ( A D ) + K ,
S ̃ ( B C ) = S ̃ ( A D ) n ext ( A B ¯ + C D ¯ ) = B C n ( x , z ) d s K ,
S ̃ ( B r C r ) = S ̃ ( A r D r ) n ext ( A r B r ¯ + C r D r ¯ ) = B r C r n ( x , z ) d s r K .
n ( r , z ) = n 0 ( z ) + n 1 ( z ) r 2 + n 2 ( z ) r 4 + = i = 0 M n i ( z ) r 2 i ,
n i ( z ) = j = 0 2 ( M i ) n i j z j
r 2 = x 2 + y 2 .
n ant ( x , z H ) = n pos ( x , z H ) ,
n ant ( x , z H ) z = n pos ( x , z H ) z .
p = 1 P r = 1 N p [ ( i = 0 M j = 0 2 ( M i ) n i j f i j p r Q K p ) S ̃ ( B p r C p r ) ] 2 ,
f i j p r Q = { B p r Q C p r Q x 2 i z j d l Q if Q = 0 q = 0 Q 1 B p r Q B p r q + 1 x 2 i z j d l q + B p r Q C p r Q x 2 i z j d l Q + q = 0 Q 1 C p r q C p r q + 1 x 2 i z j d l q if Q > 0 } .
Δ τ = B p r C p r ¯ 3 n ¯ = B p r C p r ¯ 4.2 .
Δ τ = B p r C p r ¯ ( 2 Q + 1 ) n ¯ ,
z = 0.027 x 2 ,
z = 0.064 x 2 + 3.47 .
i = 1 2 Q + 1 B p r C p r n ̂ ( x , z ) d l i S ̃ ( B p r C p r ) + K p + σ f ,
i = 1 2 Q + 1 [ B p r C p r n ( x , z ) d l i + B p r C p r Δ n ( x , z ) d l i ] S ̃ ( B p r C p r ) + Δ S ̃ ( B p r C p r ) + K p + σ f .
i = 1 2 Q + 1 B p r C p r n ( x , z ) d l i S ̃ ( B p r C p r ) + K p ,
i = 1 2 Q + 1 B p r C p r Δ n ( x , z ) d l i Δ S ̃ ( B p r C p r ) + σ f .
ϵ n ϵ S ̃ + σ f t ,
ϵ n ϵ S ̃ + σ f t n ext R eq ϵ sin + σ f t ,

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